Written in 2009.

What Is the Answer to Zeno’s Paradox? (2014), by Brian Palmer.

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Consider the blue dot at the doorway.

We know that Achilles will reach that point in exactly one second, using the numbers in our previous example.

But how, after one second, does Achilles suddenly “escape” from Zeno’s endless sequence and emerge at the doorway?

After all, only a split-second before, we find Achilles at point 85, or point 1,167, or point 345,678, depending on how fine the second is split. The finer it is split, the higher up in Zeno’s sequence we find him to be.

And yet, there, after one second, he has somehow escaped from the sequence altogether. The transition seems incomprehensible. How did he get

If the sequence had a

Something like this may underlie the essential complaint about the endlessness of Zeno’s sequence. The worry is not easy to articulate, but we can try to sharpen it further.

To see the complaint in another way, consider our infinitely-sliced pie again. Imagine

Suppose it takes ½ a second to lay down the first slice, ¼ of a second to lay down the next one, and so on, in exact proportion to size.

Then the entire pie will stand complete after exactly one second.

But this may seem baffling. Slices are being added endlessly, and suddenly, after one second, the pie stands there complete. But there was no

We can see how the pie can be completed if a

We can also see this backwards, as Zeno sometimes urges us to do.

Suppose we try to construct the pie in the reverse direction, starting with the smallest possible slice, and ending with the largest slice of 1/2.

Of course, there is no such thing as the smallest possible slice, since no matter which slice you pick, there is always a smaller one. It follows that we could never get started!

This is the same complaint, only made in reverse. Since there is no

In the same way, Zeno sometimes wonders how Achilles can ever get

Clearly, before he can get there, he must first get to point 2. But before doing so, he must obviously get to point 3. But not before he gets to point 4, and so on, without end.

It seems that Achilles cannot get started! The trouble is that there is no first point to be reached, just as there was no first slice of the pie to be laid down. Without a first point, it seems incomprehensible how Achilles can

What should we make of these considerations?

Well, before addressing them, it’s worth pausing to examine a famous view which accepts that they are correct! Thus, some people find these considerations completely persuasive and conclude that a last (or first) point in Zeno’s sequence is genuinely demanded, on pain of suffering the absurdities just mentioned.

In other words, they hold that Zeno’s sequence must somehow be “prevented” from going on endlessly in the manner so far contemplated. On their view, the correct lesson to draw is that space is not “infinitely divisible,” as Zeno implicitly assumes it to be.

This solution is easy to appreciate, so let’s consider it first.

Achilles & the tortoise

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